\documentclass{beamer}

\usepackage{amsmath,amssymb,amsthm}
\usepackage{verbatim}


\title[AMath 574 Final Project]{Penumbra Caustic Waves in Clawpack}
\author[Johnson, Swierczewski]{Mikala Johnson and Chris Swierczewski}
\date[2011]{15 March 2011}
\institute[University of Washington]{University of Washington \\ 
                                     Department of Applied Mathematics \\
                                     AMath 574 Final Talk}

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\usetheme{Warsaw}
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%\AtBeginSection[]{\begin{frame}\frametitle{Table of Contents}\tableofcontents[currentsection]\end{frame}}
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\begin{document}

\frame{\titlepage}

\frame{
  \frametitle{Table of Contents}
  \tableofcontents
}


\section{Penumbra Caustics}

\subsection{Theory}

\frame{
  \frametitle{Caustics}
  \begin{itemize}
    \item The envelope of rays either reflected or refracted by a manifold.
    \item Penumbra caustics are formed by the propagation of convex acoustic 
      wavefronts.
  \end{itemize}
  \begin{figure}
    \centering
    \includegraphics[width=0.7\textwidth]{caustics.png}
    \caption{\small Various caustics. The dark black lines are the envelopes 
      of the given set of acoustic waves.}
  \end{figure}
}

\frame{
  \frametitle{Example Penumbra Caustic: Square Root Wave}
  \begin{columns}[l]
    \column{.5\textwidth}
    \begin{itemize}
      \item Initial acoustic wave profile: $y=\sqrt{x}$.
      \item Initial velocity: normal to wave profile.
      \item No acoustic wave activity below caustic / envelope.
    \end{itemize}

    \column{.5\textwidth}
    \begin{figure}
      \centering
      \includegraphics[width=0.8\textwidth]{sqrtcaustics.png}
      \caption{\small Particle paths of the square root acoustic wavefront.}
    \end{figure}
  \end{columns}
}

\subsection{Computation}

\frame{
  \frametitle{Demo: Single Wave}
  \begin{figure}
    \centering
    \includegraphics[height=0.9\textheight]{singlenoamr07.png}
  \end{figure}
}

\subsection{Multiple Waves}

\frame{
  \frametitle{Exploring the Diffraction Pattern}
  \begin{itemize}
    \item What happens when we introduce multiple wavefronts?
    \item Diffraction pattern along the horizontal axis.
    \item Diffraction continues along the penumbra caustic.
  \end{itemize}
}

\frame{
  \frametitle{Demo: Multiple Waves}

  \begin{figure}
    \centering
    \includegraphics[height=0.9\textheight]{threewave.png}
  \end{figure}
}

\section{AMR}

\subsection{Multiple Waves}

\begin{frame}
  \frametitle{Adaptive Mesh Refinement}

  \begin{itemize}
    \item Dynamically adjust grid to accurately capture areas of high activity
    \item Cells are flagged if gradient of current solution has large 
      magnitude. Combinatorial algorithm groups cells together.
    \item Key problem: how to match across grid cells.
    \item Improve accuracy without large performance hit.
  \end{itemize}
\end{frame}

\begin{frame}[fragile]
\begin{verbatim}
code/
  Makefile
  qinit.f
  bc2.f
  ...
  setplot.py
  setrun.py
  amr/
    Makefile
    setplot.py
    setrun.py
    allowflag.f
\end{verbatim}

\end{frame}

\begin{frame}
  \frametitle{Demo: AMR}
  \begin{figure}
    \centering
    \includegraphics[width=\textwidth]{amr.png}
  \end{figure}
\end{frame}

\subsection{Experimenting with Cusp Caustics}

\frame{
  \frametitle{Example Cusp Caustic: Cube Root Wave}
  \begin{columns}[l]
    \column{.5\textwidth}
    \begin{itemize}
      \item Cusp caustics: when the caustic forms a cusp.
      \item Large amount of wave interaction.
      \item Will diffraction occur?
    \end{itemize}

    \column{.5\textwidth}
    \begin{figure}
      \centering
      \includegraphics[width=0.8\textwidth]{cubertcaustic.png}
      \caption{\small Particle paths of the cube root acoustic wavefront.}
    \end{figure}
  \end{columns}
}

\frame{
  \frametitle{Demo: Cusp Caustics}
  \begin{figure}
    \centering
    \includegraphics[height=0.9\textheight]{cusp-t12.png}
  \end{figure}
}

\begin{frame}[fragile]
  \frametitle{Thank You}
  \begin{center}
    Questions?
  \end{center}
\end{frame}
\end{document}
